ixxss2 - Intuitionistic Logic Explorer

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Theorem ixxss2 9841

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Hypotheses**
Ref	Expression
ixxssixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxss2.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})
ixxss2.3	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))

**Assertion**
Ref	Expression
ixxss2	⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝐶,𝑥,𝑦,𝑧 𝑤,𝑂,𝑥 𝑤,𝐵,𝑥,𝑦,𝑧 𝑤,𝑃 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝑥,𝑇,𝑦,𝑧 𝑤,𝑊 Allowed substitution hints: 𝑃(𝑥,𝑦,𝑧) 𝑅(𝑤) 𝑆(𝑤) 𝑇(𝑤) 𝑂(𝑦,𝑧) 𝑊(𝑥,𝑦,𝑧)

**Proof of Theorem ixxss2**
Step	Hyp	Ref	Expression
1		ixxss2.2	. . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})
2	1	elixx3g 9837	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)))
3	2	simplbi 272	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
4	3	adantl 275	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
5	4	simp3d 1001	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ ℝ^*)
6	2	simprbi 273	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
7	6	adantl 275	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
8	7	simpld 111	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴𝑅𝑤)
9	7	simprd 113	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑇𝐵)
10		simplr 520	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵𝑊𝐶)
11	4	simp2d 1000	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵 ∈ ℝ^*)
12		simpll 519	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐶 ∈ ℝ^*)
13		ixxss2.3	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
14	5, 11, 12, 13	syl3anc 1228	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
15	9, 10, 14	mp2and 430	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑆𝐶)
16	4	simp1d 999	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴 ∈ ℝ^*)
17		ixxssixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 9833	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	16, 12, 18	syl2anc 409	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 8, 15, 19	mpbir3and 1170	. . 3 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 114	. 2 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝑤 ∈ (𝐴𝑃𝐵) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3148	1 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 {crab 2448 ⊆ wss 3116 class class class wbr 3982 (class class class)co 5842 ∈ cmpo 5844 ℝ^*cxr 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937

This theorem is referenced by: iooss2 9853

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